Low pass filters are used in all branches of signal processing (video, audio, data, etc.). Their function is to pass a package of wave energy from zero frequency up to a determined cut-off frequency and reject all energy beyond this limit. An ideal low-pass filter is one that passes, without attenuation, all frequencies below the cut-off frequencies while providing infinite attenuation above the cut-off frequencies.
FIG. 1A is a frequency domain representation of a portion of a conventional input signal where a sharp transition occurs, where the x-axis represent frequency and the y-axis represents amplitude. For many applications, the input signal may be bandwidth reduced using some type of low-pass filter. However, depending on their structure, low-pass filters may react differently during the processing of a very sharp transition in the input signal, e.g., an abrupt change between black and white. For the sake of simplicity, the example signal shown is limited to video applications, but this discussion is equally applicable to all other signal sources.
In practice, no filter is ideal. All categories of low pass filters give a choice between a fast cut-off between low and high frequencies. Some low pass filters are designed with a steep cutoff slope that resembles a “Brickwall,” FIG. 1B is a frequency domain representation of the output of an ideal Brickwall-type low-pass filter, where the x-axis represent frequency and the y-axis represents frequency. Another type of low-pass filter, such as a Gaussian filter, may produce an output signal during a very sharp transition that has a more progressive variation of amplitude between high and low frequencies. FIG. 1C is a frequency domain representation of a conventional Gaussian-type low-pass filter whose output has a gradual cutoff slope.
Unfortunately, both types of low pass filters (including other filters having the same family of responses as Brick-wall filters and Gaussian filters) present artifacts or defects, particularly in the time domain in video and audio signal examples.
For example, Brickwall-type filters are known to generate excessive ringing on either side of large amplitude transitions, which is detrimental to image appearance. FIG. 1D is a time domain representation of the output of the ideal Brickwall-type low-pass filter after processing the signal of FIG. 1A, where the x-axis represent frequency and the y-axis represents time. The time domain representation shows that an advantage of the Brickwall-type filter is that it produces a very quick change between low frequencies (preserved) and high frequencies (efficiently reduced or eliminated). The inconvenience is that due to the impulse response function of the Brickwall filter, the Brickwall filter produces time domain artifacts called “ringing” in the vicinity of the cutoff area, i.e., on either side of the transition in the input signal. In the output video signal, the transitions are sharp and deliver an apparent high resolution. However, the sharp edges are surrounded by a series of “echoes” (ringing), which appear to be attenuated repetition of the sharp edges. The other family of filters, such as Gaussian filters, do not present such inconvenient ringing artifacts, but produce other problems. FIG. 1E is a frequency domain representation of a conventional Gaussian-type low-pass filter after processing the signal of FIG. 1A. The high frequencies are not sufficiently attenuated (which leads to undesirable artifacts with further processing) and, in video for example, a sudden transition in the input signal is expanded in time in the output signal (FIG. 1E), which makes the image appear to be out of focus in the vicinity of the transition.
Accordingly, there is a need for an improved low-pass filter for reducing the bandwidth of an input signal. The present invention addresses such a need.